On a class of type $II_1$ factors with Betti numbers invariants

By Sorin Popa. 2002

442 citations recorded by the source corpus.

We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A io M, the Betti numbers of the standard equivalence relation associated with A io M ([G2]), are in fact isomorphism invariants for the factors M, ¥â HT n (M), n iA 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying ¥â HT n (Mt) = ¥â HT n (M)/t, ¢£t > 0, and a Ki§unneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 SL(2,Z)), for which ¥â HT 1 (M) = ¥â1(SL(2,Z)) = 1/12. Thus, Mt M,¢£t = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R. V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0cbf498fc37fe7887a96612668b688cae61c2de1

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Citations in the corpus

Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$ (2006, methodology): “The inclusion (B ⊂ M) is rigid (or has relative property (T)) if whenever φn : M → M is a sequence of unital, tracial, completely positive (c.p.) maps such that φn →idM in the pointwise ||.||2-topology, we must have that limn→∞ supx∈B,||x||≤1 ||φn(x)−x||2 = 0 ([ P1 ]).”
A CLASS OF SUPERRIGID GROUP VON NEUMANN ALGEBRAS (2010, unknown): “, [Pop06a], [Pop06d], [IPP08], [PV08], etc.”
Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) (2006, methodology): “as. It was later reﬁned by Golodets and Nessonov [25] to obtain II1 factors with countable fundamental group containing a prescribed countable subgroup of R∗ +. However, until Popa’s breakthroughs in [49, 53, 54], no precise computation of a fundamental group diﬀerent from R∗ + had been obtained. No”
W*-superrigidity for Bernoulli actions of property (T) groups (2010, unknown): “t ||x·ξi,j−ξi,j·x|| ||ξi,j|| ≤ 3 √ ε 1−2 √ ε , for all x∈ F an every i,j∈ J. Thus, if we choose εsmall enough, then property (T) quarantees that Hi,j has a central vector, for every i6= j ∈ J ([CJ85],[Po06c]). In other words, we can ﬁnd v∈ ∆i(1)M∆j(1) such that ∆i(x)v= v∆j(x), for all x∈ M. This pro”
Drinfeld Center and Representation Theory for Monoidal Categories (2015, background): “and ε>0 such that if (H,ξ) is a pointed Hilbert bimodule over A⊂Bwith kxξ−ξxk<εfor x∈F, then there is a B-central unit vector in H. (This is not to be confused with rigidity of the inclusion A⊂B[Pop06].) For some other equivalent conditions see [Pop86, 4.1.5] and [Pop99, 9.1]. Proposition 4.12”
A random matrix approach to the Peterson-Thom conjecture (2020, background): “ments in deformation/rigidity theory go beyond free group factors and apply to von Neumann algebras associated to groups which have a combination of approximation properties and nontrivial cohomology [61, 60, 62, 63, 56, 54, 14], as well as crossed product algebras associated to actions of such grou”
Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups (2006, methodology): “We make use of another technique due to the first author, yielding unitary intertwining of subalgebras of a II1 factor by using bimodules (see [37]).”
ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, background): “This shows in particular φui(A) ≺BΛπ(i) φ(A), and since B = φ(A) and φui(A) are Cartan subalgebras of BΛπ(i), then by Theorem A2 in [31] there exists a unitary u i ∈ B Λπ(i) such that u iψu(A)u i ∗ = φ(A) = B.”
L 2-homology for von Neumann algebras (2003, methodology): “N92]) that pL(Ft)p∼=L(F ((t−1)λ−2)+1. In this respect, our theory seems to be quite disjoint from a kind of relative theory of L2- invariants for factors with HT-Cartan subalgebras discovered by Popa [Popa]. Indeed, the two theories have diﬀerent scaling properties (λ−2 versus λ−1) under compression”
The Mapping Class Group from the Viewpoint of Measure Equivalence Theory (2005, background): “to the recent solution of a longstanding problem on von Neumann algebras by Popa [88] through an excellent work of Gaboriau [28] on `(2)-Betti numbers of a measurepreserving group actions.”
Bass-Serre rigidity results in von Neumann algebras (2008, methodology): “For the purpose of this paper, we will extend the intertwining techniques from [ 32 ] as well as some results of [21] in the context of semifinite von Neumann algebras.”
Amalgamated free products of weakly rigid factors and calculation of their symmetry groups (2005, methodology): “theorems” for AFP groups in BassSerre theory. Our main “subalgebra theorem” shows that, under rather general conditions, any von Neumann subalgebra Q⊂ Mwith the relative property (T) in the sense of ([P5]) (also called a rigid inclusion), can be conjugated by an inner automorphism of Minto either M1”